## Date of Award

8-2024

## Degree Type

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Mathematics and Statistics

## First Advisor

Zhongshan Li

## Second Advisor

Marina Arav

## Third Advisor

Frank Hall

## Fourth Advisor

Hendricus van der Holst

## Abstract

Let $A$ be an $n\times n$ real matrix. As shown in the recent paper \cite{Fallat22}, if the manifolds $\mathcal{M}_A$$=\{ G^{-1}AG: G\in\text{GL}(n,\mathbb{R})\}$ and $Q($sgn$(A))$ (consisting of all real matrices having the same sign pattern as $A)$, both considered as embedded submanifolds of $\mathbb{R}^{n\times n}$, intersect transversally at $A$, then every superpattern of sgn$(A)$ also allows a matrix similar to $A.$ Such paper introduced a condition on $A$ equivalent to the above transversality, called the \emph{nonsymmetric strong spectral property} (nSSP).

In this dissertation, this transversality property of $A$ is characterized using an alternative, more direct and convenient condition, called the \emph{similarity-transversality property} (STP). Let $X=[x_{ij}]$ be a generic matrix of order $n$ whose entries are independent variables. The STP of $A$ is defned as the full row rank property of the Jacobian matrix of the entries of $AX-XA$ at the zero entry positions of $A$ with respect to the nondiagonal entries of $X.$ This new approach makes it possible to take better advantage of the combinatorial structure of the matrix $A$, and provides theoretical foundation for constructing matrices similar to a given matrix while the entries have certain desired signs. In particular, several important classes of zero-nonzero patterns and sign patterns that require or allow this transversality property are identified. Examples illustrating many possible applications are provided.

## DOI

https://doi.org/10.57709/37369282

## Recommended Citation

Yang, Zheng, "Advances on Similarity via Transversal Intersection of Manifolds." Dissertation, Georgia State University, 2024.

doi: https://doi.org/10.57709/37369282

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